When H1 and H2 are the same Hilbert space, denoted by H, the inner product⟨⋅,⋅⟩ in H is itself a bounded sesquilinear form. The boundedness condition follows from the Cauchy-Schwarz inequality.

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Let T:H1→H2 be a bounded linear operator and denote by ⟨⋅,⋅⟩ the inner product in H2. The function [⋅,⋅]:H1×H2→ℂ defined by

[ξ,η]:=⟨T⁢ξ,η⟩

is a bounded sesquilinear form. The boundedness condition follows from the Cauchy-Schwarz inequality and the fact that T is bounded.

Riesz representation of bounded sesquilinear forms

The second example above is in fact the general case. To every bounded sesquilinear form one can associate to it a unique bounded operator. That is content of the following result:

Theorem - Riesz - Let H1, H2 be two Hilbert spaces and denote by ⟨⋅,⋅⟩ the inner product in H2. For every bounded sesquilinear form [⋅,⋅]:H1×H2→ℂ there is a unique bounded linear operator T:H1→H2 such that

[ξ,η]=⟨T⁢ξ,η⟩,ξ∈H1,η∈H2.

Thus, there is a correspondence between bounded linear operators and bounded sesquilinear forms. Actually, in the early twentieth century, spectral theory was formulated solely in terms of sesquilinear forms on Hilbert spaces. Only later it was realized that this could be achieved, perhaps in a more intuitive manner, by considering linear operators instead. The linear operator approach has its advantages, as for example one can define the composition of linear operators but not of sesquilinear forms. Nevertheless it is many times useful to define a linear operator by specifying its sesquilinear form.